Phy5645/Transformations and Symmetry Problem

(a)

Given the action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}} on a state,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}|n\rangle = |n+1\rangle,}

we find that the matrix elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}} are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle i|T|j\rangle = \delta_{1,j+1}.}

We may therefore write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}} as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T} = \sum_{n=1}^{N} |n+1\rangle\langle n|,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |N+1\rangle = |1\rangle.} The Hermitian adjoint is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}^{\dagger} = \sum_{n=1}^{N} |n\rangle\langle n+1|,}

so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}\hat{T}^\dagger = \sum_{m=1}^{N}\sum_{n=1}^{N}(|m+1\rangle\langle m|)(|n\rangle\langle n+1|) = \sum_{m=1}^{N}\sum_{n=1}^{N}\delta_{m,n}|m+1\rangle\langle n+1|= \sum_{n=1}^{N}|n\rangle\langle n| = \hat{I}.}

We have thus shown that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}} is unitary.

Let us now find the commutator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}} with the Hamiltonian.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{T},\hat{H}] = \sum_{m=1}^{N}\sum_{n=1}^{N}[(|m+1\rangle\langle m|)(|n\rangle \langle n+1| + |n+1\rangle \langle n|) - (|n\rangle \langle n+1| + |n+1\rangle \langle n|)(|m+1\rangle\langle m|)]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{n=1}^{N}[|n+1\rangle\langle n+1| + |n+2\rangle\langle n| - |n\rangle\langle n| - |n+1\rangle\langle n-1|]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{n=2}^{N+1}|n\rangle\langle n| + \sum_{n=2}^{N+1}|n+1\rangle\langle n-1| - \sum_{n=1}^{N}|n\rangle\langle n| - \sum_{n=1}^{N}|n+1\rangle\langle n-1|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{n=1}^{N}[|n+2\rangle\langle n| - |n+1\rangle\langle n-1| = 0 }

Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}} commutes with the Hamiltonian.

(b)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi \rangle = \sum_{n=1}^{N} e^{ikn}|n\rangle= \sum_{n=1}^{N} e^{ikn}|n+1\rangle=\sum_{n=2}^{N+1}e^{ik(n-1)}|n\rangle=e^{-ik}\left [\sum_{n=2}^{N} e^{ikn}|n\rangle + e^{ik(N+1)}|1\rangle\right ]}

The expression in the brackets is almost Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle,} except that the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1\rangle} in general has the "wrong" coefficient. We may obtain the correct coefficient, and thus an eigenstate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T},} if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{ikN} = 1.\!}

This is satisfied if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\frac{2\pi n}{N},}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq n<N} is an integer. This range of values gives all of the unique eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T},} each with eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{-ik}.\!}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H,T} commute, any eigenstate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is an eigenstate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}

explicit proof:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{N} (|n\rangle \langle n+1| + |n+1\rangle \langle n| \sum_{m=1}^{N} e^{ikm}|m\rangle}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\sum_{n=1}^{N} \sum_{m=1}^{N} e^{ikm}|n\rangle \langle n+1|n\rangle + \sum_{n=1}^{N} \sum_{m=1}^{N} e^{ikm}|n+1\rangle \langle n|n\rangle}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 2\cos(k)|\psi\rangle}

So they are eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} also.

(c)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F|j\rangle = |n+1-j\rangle}

So Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = \sum_{n=1}^{N} |N+1-n\rangle\langle n|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^\dagger = \sum_{n=1}^{N} |n\rangle\langle N+1-n|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle FF^\dagger = \sum_{n=1}^{N} \sum_{m=1}^{N} |N+1-m\rangle\langle m||n\rangle\langle N+1-n|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{N} \sum_{m=1}^{N}|N+1-m\rangle \langle i|j \rangle \langle N+1-j|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 1}

So Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is unitary.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle FH = \sum_{m=1}^{N} |N+1-m\rangle\langle n| \sum_{n=1}^{N} (|n\rangle \langle n+1| + |n+1\rangle \langle n|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{n=1}^{N} |N+1-n\rangle \langle n-1| + \sum_{n=1}^{N} |N+1-n\rangle \langle n+1|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle HF = \sum_{n=1}^{N} (|n\rangle \langle n+1| + |n+1\rangle \langle n| \sum_{m=1}^{N} |N+1-m\rangle\langle n|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{n=1}^{N} |N+2-n\rangle \langle n| + \sum_{n=1}^{N} |N-n\rangle \langle n|}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{n=1}^{N} |N+1-n\rangle \langle n-1| + \sum_{n=1}^{N} |N+1-n\rangle \langle n+1|}

Hence they commute.

(d)

It has already been proved that F is both unitary and hermitian. Thus the eigenvalues of F are given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm 1}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F|n\rangle = |N+1-n\rangle } by intuition let us construct states given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\pm\rangle = |n\rangle \pm |N+1-n\rangle n = 1,2,3,….N/2. }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F|\psi\pm\rangle = |N+1-n\rangle \pm |N-N+n\rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \pm( |n\rangle \pm |N+1-n\rangle ) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \pm|\psi\pm\rangle }

Thus $|\psi \pm \rangle$ thus constructed are eigenstates of F. It is noted that we have $N/2$ symmetric eigenstates given by

Failed to parse (syntax error): {\displaystyle |1\rangle + |N\rangle ; |2\rangle + |N-1\rangle ; |3\rangle +|N-2\rangle … } and $N/2$ anti-symmetric eigenstates given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1\rangle - |N\rangle ; |2\rangle - |N-1\rangle ; |3\rangle -|N-2\rangle ; … }

$H(|n\rangle \pm |N+1-n\rangle )$

$=\sum _{m=1}^{N}(|m+1\rangle \langle m|+|m\rangle +\langle m+1|)(|n\rangle \pm |N+1-n\rangle )$

$=\sum _{m=1}^{N}(|m+1\rangle \langle m|n\rangle +|m\rangle \langle m+1|n\rangle \pm |m+1\rangle \langle m|N+1-n\rangle \pm |m\rangle \langle m+1|N+1-n\rangle )$

$=|n+1\rangle +|n-1\rangle \pm |N+2-n\rangle \pm |N-n\rangle$

Thus we find that the eigenstate of F may not necessarily be eigenstates oh H. This is because F is degenerate. However there may exist linear superposition of eigenstates of F which is also an eigenstates of H and vice versa.

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